Algebraic algebra

An algebra with associative powers (in particular, an associative algebra) over a field all elements of which are algebraic (an element $a$ of the algebra $A$ is called algebraic over the field $F$ if the subalgebra $F[a]$ generated by $a$ is finite-dimensional or, equivalently, if the element $a$ has an annihilating polynomial with coefficients from the ground field $F$). An algebra $A$ is called an algebraic algebra of bounded degree if it is algebraic and if the set of degrees of the minimal annihilating polynomials of its elements is bounded. Subalgebras and homomorphic images of an algebraic algebra (of bounded degree) are algebraic algebras (of bounded degree).

Examples: locally finite algebras (in particular, finite-dimensional ones), nil algebras and associative skew-fields with a countable set of generators over an uncountable field.

The algebras considered below are associative. The Jacobson radical of an algebraic algebra is a nil ideal. A primitive algebraic algebra $A$ is isomorphic to a dense algebra of linear transformations of a vector space over a skew-field; if, in addition, $A$ is of bounded degree, then $A$ is isomorphic to a ring of matrices over a skew-field. An algebraic algebra without non-zero nilpotent elements (in particular, a skew-field) over a finite field is commutative. It follows that finite skew-fields are commutative. An algebraic algebra of bounded degree satisfies a polynomial identity (cf. PI-algebra). An algebraic PI-algebra is locally finite. If the ground field is uncountable, then the algebras obtained from an algebraic algebra by extension of the ground field, and the tensor product of algebraic algebras, are algebraic algebras.

References

[1]	N. Jacobson, "Structure of rings" , Amer. Math. Soc. (1956)
[2]	I.N. Herstein, "Noncommutative rings" , Math. Assoc. Amer. (1968)